# Properties

 Label 768.4.a.q Level $768$ Weight $4$ Character orbit 768.a Self dual yes Analytic conductor $45.313$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,4,Mod(1,768)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("768.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 768.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$45.3134668844$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.1436.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 11x - 12$$ x^3 - 11*x - 12 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{5}$$ Twist minimal: no (minimal twist has level 24) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 3 q^{3} + ( - \beta_{2} - 3) q^{5} + ( - \beta_1 + 5) q^{7} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 + (-b2 - 3) * q^5 + (-b1 + 5) * q^7 + 9 * q^9 $$q - 3 q^{3} + ( - \beta_{2} - 3) q^{5} + ( - \beta_1 + 5) q^{7} + 9 q^{9} + ( - 2 \beta_{2} + 2 \beta_1) q^{11} + (3 \beta_{2} - \beta_1 - 18) q^{13} + (3 \beta_{2} + 9) q^{15} + (6 \beta_{2} + 2 \beta_1 + 6) q^{17} + (6 \beta_{2} + 2 \beta_1 - 12) q^{19} + (3 \beta_1 - 15) q^{21} + (2 \beta_1 + 54) q^{23} + (12 \beta_{2} - 4 \beta_1 + 15) q^{25} - 27 q^{27} + ( - 5 \beta_{2} + 2 \beta_1 - 57) q^{29} + ( - 12 \beta_{2} + 3 \beta_1 + 109) q^{31} + (6 \beta_{2} - 6 \beta_1) q^{33} + ( - 14 \beta_{2} - 2 \beta_1 + 36) q^{35} + (3 \beta_{2} - 11 \beta_1 - 96) q^{37} + ( - 9 \beta_{2} + 3 \beta_1 + 54) q^{39} + ( - 6 \beta_{2} + 14 \beta_1 - 42) q^{41} + (18 \beta_{2} - 10 \beta_1 + 84) q^{43} + ( - 9 \beta_{2} - 27) q^{45} + (12 \beta_{2} - 6 \beta_1 + 66) q^{47} + ( - 24 \beta_{2} + 117) q^{49} + ( - 18 \beta_{2} - 6 \beta_1 - 18) q^{51} + (11 \beta_{2} + 10 \beta_1 - 369) q^{53} + (36 \beta_{2} - 4 \beta_1 + 160) q^{55} + ( - 18 \beta_{2} - 6 \beta_1 + 36) q^{57} + ( - 8 \beta_{2} - 8 \beta_1 - 60) q^{59} + (3 \beta_{2} + 9 \beta_1 - 516) q^{61} + ( - 9 \beta_1 + 45) q^{63} + ( - 18 \beta_{2} + 10 \beta_1 - 288) q^{65} + (48 \beta_{2} + 204) q^{67} + ( - 6 \beta_1 - 162) q^{69} + ( - 36 \beta_{2} - 6 \beta_1 - 270) q^{71} + ( - 24 \beta_{2} - 16 \beta_1 - 146) q^{73} + ( - 36 \beta_{2} + 12 \beta_1 - 45) q^{75} + (20 \beta_{2} - 20 \beta_1 - 768) q^{77} + (12 \beta_{2} + 7 \beta_1 + 1) q^{79} + 81 q^{81} + (14 \beta_{2} + 2 \beta_1 - 384) q^{83} + ( - 42 \beta_{2} + 28 \beta_1 - 906) q^{85} + (15 \beta_{2} - 6 \beta_1 + 171) q^{87} + ( - 12 \beta_{2} - 4 \beta_1 + 42) q^{89} + (18 \beta_{2} + 38 \beta_1 + 192) q^{91} + (36 \beta_{2} - 9 \beta_1 - 327) q^{93} + ( - 24 \beta_{2} + 28 \beta_1 - 852) q^{95} + (36 \beta_{2} - 4 \beta_1 - 418) q^{97} + ( - 18 \beta_{2} + 18 \beta_1) q^{99}+O(q^{100})$$ q - 3 * q^3 + (-b2 - 3) * q^5 + (-b1 + 5) * q^7 + 9 * q^9 + (-2*b2 + 2*b1) * q^11 + (3*b2 - b1 - 18) * q^13 + (3*b2 + 9) * q^15 + (6*b2 + 2*b1 + 6) * q^17 + (6*b2 + 2*b1 - 12) * q^19 + (3*b1 - 15) * q^21 + (2*b1 + 54) * q^23 + (12*b2 - 4*b1 + 15) * q^25 - 27 * q^27 + (-5*b2 + 2*b1 - 57) * q^29 + (-12*b2 + 3*b1 + 109) * q^31 + (6*b2 - 6*b1) * q^33 + (-14*b2 - 2*b1 + 36) * q^35 + (3*b2 - 11*b1 - 96) * q^37 + (-9*b2 + 3*b1 + 54) * q^39 + (-6*b2 + 14*b1 - 42) * q^41 + (18*b2 - 10*b1 + 84) * q^43 + (-9*b2 - 27) * q^45 + (12*b2 - 6*b1 + 66) * q^47 + (-24*b2 + 117) * q^49 + (-18*b2 - 6*b1 - 18) * q^51 + (11*b2 + 10*b1 - 369) * q^53 + (36*b2 - 4*b1 + 160) * q^55 + (-18*b2 - 6*b1 + 36) * q^57 + (-8*b2 - 8*b1 - 60) * q^59 + (3*b2 + 9*b1 - 516) * q^61 + (-9*b1 + 45) * q^63 + (-18*b2 + 10*b1 - 288) * q^65 + (48*b2 + 204) * q^67 + (-6*b1 - 162) * q^69 + (-36*b2 - 6*b1 - 270) * q^71 + (-24*b2 - 16*b1 - 146) * q^73 + (-36*b2 + 12*b1 - 45) * q^75 + (20*b2 - 20*b1 - 768) * q^77 + (12*b2 + 7*b1 + 1) * q^79 + 81 * q^81 + (14*b2 + 2*b1 - 384) * q^83 + (-42*b2 + 28*b1 - 906) * q^85 + (15*b2 - 6*b1 + 171) * q^87 + (-12*b2 - 4*b1 + 42) * q^89 + (18*b2 + 38*b1 + 192) * q^91 + (36*b2 - 9*b1 - 327) * q^93 + (-24*b2 + 28*b1 - 852) * q^95 + (36*b2 - 4*b1 - 418) * q^97 + (-18*b2 + 18*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 9 q^{3} - 10 q^{5} + 14 q^{7} + 27 q^{9}+O(q^{10})$$ 3 * q - 9 * q^3 - 10 * q^5 + 14 * q^7 + 27 * q^9 $$3 q - 9 q^{3} - 10 q^{5} + 14 q^{7} + 27 q^{9} - 52 q^{13} + 30 q^{15} + 26 q^{17} - 28 q^{19} - 42 q^{21} + 164 q^{23} + 53 q^{25} - 81 q^{27} - 174 q^{29} + 318 q^{31} + 92 q^{35} - 296 q^{37} + 156 q^{39} - 118 q^{41} + 260 q^{43} - 90 q^{45} + 204 q^{47} + 327 q^{49} - 78 q^{51} - 1086 q^{53} + 512 q^{55} + 84 q^{57} - 196 q^{59} - 1536 q^{61} + 126 q^{63} - 872 q^{65} + 660 q^{67} - 492 q^{69} - 852 q^{71} - 478 q^{73} - 159 q^{75} - 2304 q^{77} + 22 q^{79} + 243 q^{81} - 1136 q^{83} - 2732 q^{85} + 522 q^{87} + 110 q^{89} + 632 q^{91} - 954 q^{93} - 2552 q^{95} - 1222 q^{97}+O(q^{100})$$ 3 * q - 9 * q^3 - 10 * q^5 + 14 * q^7 + 27 * q^9 - 52 * q^13 + 30 * q^15 + 26 * q^17 - 28 * q^19 - 42 * q^21 + 164 * q^23 + 53 * q^25 - 81 * q^27 - 174 * q^29 + 318 * q^31 + 92 * q^35 - 296 * q^37 + 156 * q^39 - 118 * q^41 + 260 * q^43 - 90 * q^45 + 204 * q^47 + 327 * q^49 - 78 * q^51 - 1086 * q^53 + 512 * q^55 + 84 * q^57 - 196 * q^59 - 1536 * q^61 + 126 * q^63 - 872 * q^65 + 660 * q^67 - 492 * q^69 - 852 * q^71 - 478 * q^73 - 159 * q^75 - 2304 * q^77 + 22 * q^79 + 243 * q^81 - 1136 * q^83 - 2732 * q^85 + 522 * q^87 + 110 * q^89 + 632 * q^91 - 954 * q^93 - 2552 * q^95 - 1222 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 11x - 12$$ :

 $$\beta_{1}$$ $$=$$ $$4\nu^{2} - 29$$ 4*v^2 - 29 $$\beta_{2}$$ $$=$$ $$4\nu^{2} - 8\nu - 29$$ 4*v^2 - 8*v - 29
 $$\nu$$ $$=$$ $$( -\beta_{2} + \beta_1 ) / 8$$ (-b2 + b1) / 8 $$\nu^{2}$$ $$=$$ $$( \beta _1 + 29 ) / 4$$ (b1 + 29) / 4

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.48361 3.76644 −1.28282
0 −3.00000 0 −18.5422 0 9.32669 0 9.00000 0
1.2 0 −3.00000 0 −0.612661 0 −22.7441 0 9.00000 0
1.3 0 −3.00000 0 9.15486 0 27.4175 0 9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.4.a.q 3
3.b odd 2 1 2304.4.a.bw 3
4.b odd 2 1 768.4.a.s 3
8.b even 2 1 768.4.a.t 3
8.d odd 2 1 768.4.a.r 3
12.b even 2 1 2304.4.a.bv 3
16.e even 4 2 96.4.d.a 6
16.f odd 4 2 24.4.d.a 6
24.f even 2 1 2304.4.a.bt 3
24.h odd 2 1 2304.4.a.bu 3
48.i odd 4 2 288.4.d.d 6
48.k even 4 2 72.4.d.d 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.4.d.a 6 16.f odd 4 2
72.4.d.d 6 48.k even 4 2
96.4.d.a 6 16.e even 4 2
288.4.d.d 6 48.i odd 4 2
768.4.a.q 3 1.a even 1 1 trivial
768.4.a.r 3 8.d odd 2 1
768.4.a.s 3 4.b odd 2 1
768.4.a.t 3 8.b even 2 1
2304.4.a.bt 3 24.f even 2 1
2304.4.a.bu 3 24.h odd 2 1
2304.4.a.bv 3 12.b even 2 1
2304.4.a.bw 3 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(768))$$:

 $$T_{5}^{3} + 10T_{5}^{2} - 164T_{5} - 104$$ T5^3 + 10*T5^2 - 164*T5 - 104 $$T_{7}^{3} - 14T_{7}^{2} - 580T_{7} + 5816$$ T7^3 - 14*T7^2 - 580*T7 + 5816 $$T_{11}^{3} - 2816T_{11} - 49152$$ T11^3 - 2816*T11 - 49152 $$T_{19}^{3} + 28T_{19}^{2} - 11088T_{19} + 274752$$ T19^3 + 28*T19^2 - 11088*T19 + 274752

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$(T + 3)^{3}$$
$5$ $$T^{3} + 10 T^{2} - 164 T - 104$$
$7$ $$T^{3} - 14 T^{2} - 580 T + 5816$$
$11$ $$T^{3} - 2816T - 49152$$
$13$ $$T^{3} + 52 T^{2} - 1104 T - 55872$$
$17$ $$T^{3} - 26 T^{2} - 11124 T + 477576$$
$19$ $$T^{3} + 28 T^{2} - 11088 T + 274752$$
$23$ $$T^{3} - 164 T^{2} + 6384 T - 45504$$
$29$ $$T^{3} + 174 T^{2} + 3964 T - 61368$$
$31$ $$T^{3} - 318 T^{2} + 4476 T + 3749624$$
$37$ $$T^{3} + 296 T^{2} - 46080 T - 82944$$
$41$ $$T^{3} + 118 T^{2} + \cdots - 19985976$$
$43$ $$T^{3} - 260 T^{2} - 80976 T + 8601408$$
$47$ $$T^{3} - 204 T^{2} - 27792 T + 1964736$$
$53$ $$T^{3} + 1086 T^{2} + \cdots + 20665224$$
$59$ $$T^{3} + 196 T^{2} - 50000 T - 8523584$$
$61$ $$T^{3} + 1536 T^{2} + \cdots + 104841216$$
$67$ $$T^{3} - 660 T^{2} + \cdots + 32228928$$
$71$ $$T^{3} + 852 T^{2} + \cdots - 85084992$$
$73$ $$T^{3} + 478 T^{2} + \cdots - 120833304$$
$79$ $$T^{3} - 22 T^{2} - 71524 T + 7902616$$
$83$ $$T^{3} + 1136 T^{2} + \cdots + 37953536$$
$89$ $$T^{3} - 110 T^{2} - 41364 T - 1423656$$
$97$ $$T^{3} + 1222 T^{2} + \cdots - 74802424$$